Sure, but between "all edges have positive weights" and "there are cycles with negative total weight" there are still graphs that have negative edge weights, ye

Sure, but between "all edges have positive weights" and "there are cycles with negative total weight" there are still graphs that have negative edge weights, yet there's no negative-weight cycle. Example: V = { A, B, C } E = { A -> B, B -> C, C -> A } weight(A -> B) = 1 weight(B -> C) = 1 weight(C -> A) = -1